It can fire it and it will shoot but if you continue it will melt the barrel

You'd have to shoot A LOT of 556 to warp the barrel... A LOT, without allowing the barrel to cool. The same thing can happen of you send a lot of 223 down range in your 223 chambered firearm in rapid succession with out a cool down period.

The issue with running 556 in a 223 chambered firearm is that the 556 runs at higher pressure. The added stress could lead to failures such as cracking you gas block and gas tube, failure to eject, failure to extract, etc.